Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
12. lim (x → ∞) (x - 8x²) / (12x² + 5x)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.2.64
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
64. y = 1/(t(t+1)(t+2))
Verified step by step guidance1
Start by expressing the function clearly: \(y = \frac{1}{t(t+1)(t+2)}\). This can also be written as \(y = [t(t+1)(t+2)]^{-1}\) to prepare for logarithmic differentiation.
Take the natural logarithm of both sides to simplify the product inside the logarithm: \(\ln y = \ln \left( [t(t+1)(t+2)]^{-1} \right)\).
Use the logarithm power rule to bring down the exponent: \(\ln y = -1 \cdot \ln [t(t+1)(t+2)]\).
Apply the logarithm product rule to expand the right side: \(\ln y = - (\ln t + \ln (t+1) + \ln (t+2))\).
Differentiate both sides with respect to \(t\). Remember that \(\frac{d}{dt} (\ln y) = \frac{1}{y} \frac{dy}{dt}\) by the chain rule, and differentiate each logarithmic term on the right side accordingly.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithmic Differentiation
Logarithmic differentiation is a technique used to differentiate functions that are products, quotients, or powers of variables by taking the natural logarithm of both sides. This simplifies the differentiation process by converting multiplication into addition and division into subtraction, making it easier to handle complex expressions.
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Logarithmic Differentiation
Properties of Logarithms
The properties of logarithms, such as log(ab) = log a + log b and log(a/b) = log a - log b, allow us to break down complicated products and quotients into sums and differences. These properties are essential in logarithmic differentiation to simplify the function before differentiating.
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Implicit Differentiation
Implicit differentiation involves differentiating both sides of an equation with respect to the independent variable, treating the dependent variable as a function of that variable. In logarithmic differentiation, after taking logs, implicit differentiation is used to find dy/dx by differentiating the transformed equation.
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Related Practice
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
33. y=ln(arctan(x))
Textbook Question
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
61. y = √(θ + 3) sin θ
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
51. lim (θ → 0) (θ - sin θ cos θ) / (tan θ - θ)
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Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
29. y=arcsec(1/t), 0<t<1
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
27. y=arccsc(x²+1)